Translation of 0-dimensional objects

Question

As far as I can understand we can think of any line on a plane as a product of translation of some other arbitrary line by some arbitrary vector (and vice versa). But while it makes sense when we talk about lines, I cannot imagine the same procedure for dots. Is it even "possible" to translate a dot such that its origin and a translated "copy" would define a line?

Answer 1

Any point is a translate of any other point: $$y = x + (y - x).$$

Taking any two points $x_0\ne y_0$ of a line, the line can be written as the set $$\{x_0 + t(y_0 - x_0)\,\vert\,t\in\Bbb R\}.$$